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Discrete probability distribution From Wikipedia, the free encyclopedia

In probability theory and statistics, the **Poisson distribution** (/ˈpwɑːsɒn/; French pronunciation: [pwasɔ̃]) is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known constant mean rate and independently of the time since the last event.^{[1]} It can also be used for the number of events in other types of intervals than time, and in dimension greater than 1 (e.g., number of events in a given area or volume).

Probability mass function | |||

Cumulative distribution function | |||

Notation | |||
---|---|---|---|

Parameters | (rate) | ||

Support | (Natural numbers starting from 0) | ||

PMF | |||

CDF |
or or (for where is the upper incomplete gamma function, is the floor function, and is the regularized gamma function) | ||

Mean | |||

Median | |||

Mode | |||

Variance | |||

Skewness | |||

Excess kurtosis | |||

Entropy |
or for large | ||

MGF | |||

CF | |||

PGF | |||

Fisher information |

The Poisson distribution is named after French mathematician Siméon Denis Poisson. It plays an important role for discrete-stable distributions.

Under a Poisson distribution with the expectation of *λ* events in a given interval, the probability of *k* events in the same interval is:^{[2]}^{: 60 }

For instance, consider a call center which receives an average of *λ =* 3 calls per minute at all times of day. If the calls are independent, receiving one does not change the probability of when the next one will arrive. Under these assumptions, the number *k* of calls received during any minute has a Poisson probability distribution. Receiving *k =* 1 to 4 calls then has a probability of about 0.77, while receiving 0 or at least 5 calls has a probability of about 0.23.

A classic example used to motivate the Poisson distribution is the number of radioactive decay events during a fixed observation period.^{[3]}

The distribution was first introduced by Siméon Denis Poisson (1781–1840) and published together with his probability theory in his work *Recherches sur la probabilité des jugements en matière criminelle et en matière civile* (1837).^{[4]}^{: 205-207 } The work theorized about the number of wrongful convictions in a given country by focusing on certain random variables N that count, among other things, the number of discrete occurrences (sometimes called "events" or "arrivals") that take place during a time-interval of given length. The result had already been given in 1711 by Abraham de Moivre in *De Mensura Sortis seu; de Probabilitate Eventuum in Ludis a Casu Fortuito Pendentibus* .^{[5]}^{: 219 }^{[6]}^{: 14-15 }^{[7]}^{: 193 }^{[8]}^{: 157 } This makes it an example of Stigler's law and it has prompted some authors to argue that the Poisson distribution should bear the name of de Moivre.^{[9]}^{[10]}

In 1860, Simon Newcomb fitted the Poisson distribution to the number of stars found in a unit of space.^{[11]}
A further practical application was made by Ladislaus Bortkiewicz in 1898. Bortkiewicz showed that the frequency with which soldiers in the Prussian army were accidentally killed by horse kicks could be well modeled by a Poisson distribution.^{[12]}^{: 23-25 }.

A discrete random variable X is said to have a Poisson distribution with parameter if it has a probability mass function given by:^{[2]}^{: 60 }

where

- k is the number of occurrences ()
- e is Euler's number ()
*k*! =*k*(*k–*1) ··· (3)(2)(1) is the factorial.

The positive real number λ is equal to the expected value of X and also to its variance.^{[13]}

The Poisson distribution can be applied to systems with a large number of possible events, each of which is rare. The number of such events that occur during a fixed time interval is, under the right circumstances, a random number with a Poisson distribution.

The equation can be adapted if, instead of the average number of events we are given the average rate at which events occur. Then and:^{[14]}

The Poisson distribution may be useful to model events such as:

- the number of meteorites greater than 1-meter diameter that strike Earth in a year;
- the number of laser photons hitting a detector in a particular time interval;
- the number of students achieving a low and high mark in an exam; and
- locations of defects and dislocations in materials.

Examples of the occurrence of random points in space are: the locations of asteroid impacts with earth (2-dimensional), the locations of imperfections in a material (3-dimensional), and the locations of trees in a forest (2-dimensional).^{[15]}

The Poisson distribution is an appropriate model if the following assumptions are true:

- k, a nonnegative integer, is the number of times an event occurs in an interval.
- The occurrence of one event does not affect the probability of a second event.
- The average rate at which events occur is independent of any occurrences.
- Two events cannot occur at exactly the same instant.

If these conditions are true, then k is a Poisson random variable; the distribution of k is a Poisson distribution.

The Poisson distribution is also the limit of a binomial distribution, for which the probability of success for each trial equals λ divided by the number of trials, as the number of trials approaches infinity (see Related distributions).

On a particular river, overflow floods occur once every 100 years on average. Calculate the probability of k = 0, 1, 2, 3, 4, 5, or 6 overflow floods in a 100-year interval, assuming the Poisson model is appropriate. Because the average event rate is one overflow flood per 100 years, λ = 1 |
k P(k overflow floods in 100 years) 0 0.368 1 0.368 2 0.184 3 0.061 4 0.015 5 0.003 6 0.0005
The probability for 0 to 6 overflow floods in a 100-year period. |